Helical solenoid trajectory with zero incident angular momentum

Let a particle enter a uniform solenoid at transverse position $u_0=x_0+i{y}_0$ with zero radial velocity after an ideal fringe. Define signed $\Omega =qB/(2\gamma m)$ and initial field-free speed $v$. Its Lorentz force trajectory inside isThe transverse motion is a circle about the guiding centre $u_0/2$ at cyclotron angular frequency $2|\Omega |$. Cylindrical radius is $|u_0\cos \Omega t|$; its apparent azimuthal jumps when the orbit crosses the axis are coordinate singularities. Forward entry requires $v>|\Omega u_0|$ in this ideal model.